University of Cambridge > > Isaac Newton Institute Seminar Series > Isotonic regression in general dimensions

Isotonic regression in general dimensions

Add to your list(s) Download to your calendar using vCal

If you have a question about this talk, please contact

STSW02 - Statistics of geometric features and new data types

Co-authors: Qiyang Han (University of Washington), Tengyao Wang (University of Cambridge), Sabyasachi Chatterjee (University of Illinois)

We study the least squares regression function estimator over the class of real-valued functions on $[0,1]d$ that are increasing in each coordinate. For uniformly bounded signals and with a fixed, cubic lattice design, we establish that the estimator achieves the minimax rate of order $n{−min\{2/(d+2),1/d\}}$ in the empirical $L_2$ loss, up to poly-logarithmic factors. Further, we prove a sharp oracle inequality, which reveals in particular that when the true regression function is piecewise constant on $k$ hyperrectangles, the least squares estimator enjoys a faster, adaptive rate of convergence of $(k/n)^{min(1,2/d)}$, again up to poly-logarithmic factors. Previous results are confined to the case $d\leq 2$. Finally, we establish corresponding bounds (which are new even in the case $d=2$) in the more challenging random design setting. There are two surprising features of these results: first, they demonstrate that it is possible for a global empirical risk minimisation procedure to be rate optimal up to poly-logarithmic factors even when the corresponding entropy integral for the function class diverges rapidly; second, they indicate that the adaptation rate for shape-constrained estimators can be strictly worse than the parametric rate. 

This talk is part of the Isaac Newton Institute Seminar Series series.

Tell a friend about this talk:

This talk is included in these lists:

Note that ex-directory lists are not shown.


© 2006-2018, University of Cambridge. Contact Us | Help and Documentation | Privacy and Publicity